Eigenmode analysis of the multiple temperature model: spectrum properties, hierarchical structures, and temperature inversion

• Published in: Applied physics. A, Materials science & processing Materials science & processing, 2023-03, Vol.129 (3), Article 165
• Main Authors: Katow, Hiroki, Ishikawa, Kenichi L.
• Format: Article
• Language: English
• Subjects: Applied physics; Characterization and Evaluation of Materials; Condensed Matter Physics; Electron spin; Exact solutions; Fourier series; Green's functions; Machines; Manufacturing; Materials science; Nanotechnology; Optical and Electronic Materials; Physics; Physics and Astronomy; Processes; Series expansion; Subsystems; Surfaces and Interfaces; Thin Films; Ultrafast lasers
• ISSN: 0947-8396; 1432-0630
• DOI: 10.1007/s00339-023-06429-z

Recent developments of ultrafast laser pulse techniques enable us to study the subpicosecond scale dynamics out of thermal equilibrium. Multiple temperature models (MTMs) are frequently used to describe such dynamics where the total system is divided into subsystems each of which is in local thermal equilibrium. Typical examples include the electron-lattice two temperature model and electron-spin-phonon three temperature model. We present the exact analytical solutions of linear MTM, based on the Fourier series expansion and the Green’s function method. We then discuss their properties for the case of the two and three temperature models. We show that the general solution of MTM is expressed as a linear combinations of a spatially uniform, single-temperature stationary mode and the other non-oscillatory, decaying “eigenmodes” characterized by different wave vectors and well-defined mode lifetimes. The eigenmode picture enables us to explore the hierarchical structure of models with respect to space, time and the coupling parameter. Excitation by source term is included by the Green’s function method. As an example, we derive an analytical solution for a Gaussian type source term. We report a phenomenon “temperature inversion” where the lattice temperature exceeds electron’s temperature for ns time scale. Furthermore, we show how physical requirements such as energy conservation and equilibration are realized in the general linear MTM in terms of the eigenmode picture.